Consistency Conditions on S-Matrix of Spin 1 Massless Particles

Motivated by new techniques in the computation of scattering amplitudes of massless particles in four dimensions, like BCFW recursion relations, the question of how much structure of the S-matrix can be determined from purely S-matrix arguments has received new attention. The BCFW recursion relations for massless particles of spin 1 and 2 imply that the whole tree-level S-matrix can be determined in terms of three-particle amplitudes (evaluated at complex momenta). However, the known proofs of the validity of the relations rely on the Lagrangian of the theory, either by using Feynman diagrams explicitly or by studying the effective theory at large complex momenta. This means that a purely S-matrix theoretic proof of the relations is still missing. The aim of this paper is to provide such a proof for spin 1 particles by extending the four-particle test introduced by P. Benincasa and F. Cachazo in arXiv:0705.4305[hep-th] to all particles. We show how n-particle tests imply that the rational function built from the BCFW recursion relations possesses all the correct factorization channels including holomorphic and anti-holomorphic collinear limits. This in turn implies that they give the correct S-matrix of the theory.Comment: 24 pages, 4 figure

The Yangian origin of the Grassmannian integral

In this paper we analyse formulas which reproduce different contributions to scattering amplitudes in N=4 super Yang-Mills theory through a Grassmannian integral. Recently their Yangian invariance has been proved directly by using the explicit expression of the Yangian level-one generators. The specific cyclic structure of the form integrated over the Grassmannian enters in a crucial way in demonstrating the symmetry. Here we show that the Yangian symmetry fixes this structure uniquely.Comment: 26 pages. v2: typos corrected, published versio

From lightcone actions to maximally supersymmetric amplitudes

In this article actions for N=4 SYM and N=8 supergravity are formulated in terms of a chiral superfield, which contains only the physical degrees of freedom of either theory. In these new actions, which originate from the lightcone superspace, the supergravity cubic vertex is the square of the gauge theory one (omitting the color structures). Amplitude calculations using the corresponding Feynman supergraph rules are tedious, but can be simplified by choosing a preferred superframe. Recursive calculations of all MHV amplitudes in N=4 SYM and the four-point N=8 supergravity amplitude are shown to agree with the known results and connections to the BCFW recursion relations are pointed out. Finally, the new path integrals are discussed in the context of the double-copy property relating N=4 SYM theory to N=8 supergravity.Comment: 29 pages, 2 figures, v2: title modified, published versio

Dual conformal constraints and infrared equations from global residue theorems in N=4 SYM theory

Infrared equations and dual conformal constraints arise as consistency conditions on loop amplitudes in N=4 super Yang-Mills theory. These conditions are linear relations between leading singularities, which can be computed in the Grassmannian formulation of N=4 super Yang-Mills theory proposed recently. Examples for infrared equations have been shown to be implied by global residue theorems in the Grassmannian picture. Both dual conformal constraints and infrared equations are mapped explicitly to global residue theorems for one-loop next-to-maximally-helicity-violating amplitudes. In addition, the identity relating the BCFW and its parity-conjugated form of tree-level amplitudes, is shown to emerge from a particular combination of global residue theorems.Comment: 21 page

The Grassmannian and the Twistor String: Connecting All Trees in N=4 SYM

We present a new, explicit formula for all tree-level amplitudes in N=4 super Yang-Mills. The formula is written as a certain contour integral of the connected prescription of Witten's twistor string, expressed in link variables. A very simple deformation of the integrand gives directly the Grassmannian integrand proposed by Arkani-Hamed et al. together with the explicit contour of integration. The integral is derived by iteratively adding particles to the Grassmannian integral, one particle at a time, and makes manifest both parity and soft limits. The formula is shown to be related to those given by Dolan and Goddard, and generalizes the results of earlier work for NMHV and N^2MHV to all N^(k-2)MHV tree amplitudes in N=4 super Yang-Mills.Comment: 26 page

A note on the boundary contribution with bad deformation in gauge theory

Motivated by recently progresses in the study of BCFW recursion relation with nonzero boundary contributions for theories with scalars and fermions\cite{Bofeng}, in this short note we continue the study of boundary contributions of gauge theory with the bad deformation. Unlike cases with scalars or fermions, it is hard to use Feynman diagrams directly to obtain boundary contributions, thus we propose another method based on the ${\cal N}=4$ SYM theory. Using this method, we are able to write down a useful on-shell recursion relation to calculate boundary contributions from related theories. Our result shows the cut-constructibility of gauge theory even with the bad deformation in some generalized sense.Comment: 16 pages, 7 figure

On All-loop Integrands of Scattering Amplitudes in Planar N=4 SYM

We study the relationship between the momentum twistor MHV vertex expansion of planar amplitudes in N=4 super-Yang-Mills and the all-loop generalization of the BCFW recursion relations. We demonstrate explicitly in several examples that the MHV vertex expressions for tree-level amplitudes and loop integrands satisfy the recursion relations. Furthermore, we introduce a rewriting of the MHV expansion in terms of sums over non-crossing partitions and show that this cyclically invariant formula satisfies the recursion relations for all numbers of legs and all loop orders.Comment: 34 pages, 17 figures; v2: Minor improvements to exposition and discussion, updated references, typos fixe

Dual Conformal Properties of Six-Dimensional Maximal Super Yang-Mills Amplitudes

We demonstrate that the tree-level amplitudes of maximal super-Yang-Mills theory in six dimensions, when stripped of their overall momentum and supermomentum delta functions, are covariant with respect to the six-dimensional dual conformal group. Using the generalized unitarity method, we demonstrate that this property is also present for loop amplitudes. Since the six-dimensional amplitudes can be interpreted as massive four-dimensional ones, this implies that the six-dimensional symmetry is also present in the massively regulated four-dimensional maximal super-Yang-Mills amplitudes.Comment: 20 pages, 3 figures, minor clarification, references update

No triangles on the moduli space of maximally supersymmetric gauge theory

Maximally supersymmetric gauge theory in four dimensions has a remarkably simple S-matrix at the origin of its moduli space at both tree and loop level. This leads to the question what, if any, of this structure survives at the complement of this one point. Here this question is studied in detail at one loop for the branch of the moduli space parameterized by a vacuum expectation value for one complex scalar. Motivated by the parallel D-brane picture of spontaneous symmetry breaking a simple relation is demonstrated between the Lagrangian of broken super Yang-Mills theory and that of its higher dimensional unbroken cousin. Using this relation it is proven both through an on- as well as an off-shell method there are no so-called triangle coefficients in the natural basis of one-loop functions at any finite point of the moduli space for the theory under study. The off-shell method yields in addition absence of rational terms in a class of theories on the Coulomb branch which includes the special case of maximal supersymmetry. The results in this article provide direct field theory evidence for a recently proposed exact dual conformal symmetry motivated by the AdS/CFT correspondence.Comment: 39 pages, 4 figure

Spinor Helicity and Dual Conformal Symmetry in Ten Dimensions

The spinor helicity formalism in four dimensions has become a very useful tool both for understanding the structure of amplitudes and also for practical numerical computation of amplitudes. Recently, there has been some discussion of an extension of this formalism to higher dimensions. We describe a particular implementation of the spinor-helicity method in ten dimensions. Using this tool, we study the tree-level S-matrix of ten dimensional super Yang-Mills theory, and prove that the theory enjoys a dual conformal symmetry. Implications for four-dimensional computations are discussed.Comment: 24 pages, 1 figure